In The Matrix, when Smith meets Neo, he calls him "Mr Anderson", which is his Matrix name.
Of course people born in the real world chose their own names, their real names.
The hackers who become aware of The Matrix and are freed go by heir hacker names once out.
Hence Neo, Trinity, etc. Their Matrix names are assigned to them at birth by the machines, along with an identity that never really existed.
Smith makes a point of calling Neo "Mr Anderson" every time they meet. Neo eventually makes the point of saying "my name is Neo" in response.
When Smith meets Cypher he calls him "Mr Regan" - his Matrix name.
Yet when he finally meets Morpheus he calls him "The Great Morpheus", rather than use his Matrix name (which we never do find out). Even under interogation, Smith never calls him "Mr Jones" or whatever.
Oversight or deliberate?